Full rotational symmetry from reflections or rotational symmetries in finitely many subspaces

Two related questions are discussed. The first is when reflection symmetry in a finite set of i-dimensional subspaces, i ∈ {1, . . . , n − 1}, implies full rotational symmetry, that is, the closure of the group generated by the reflections equals O(n). For i = n − 1, this has essentially been solved by Burchard, Chambers, and Dranovski, but new results are obtained for i ∈ {1, . . . , n−2}. The second question, to which an essentially complete answer is given, is when (full) rotational symmetry with respect to a finite set of i-dimensional subspaces, i ∈ {1, . . . , n − 2}, implies full ro- tational symmetry, that is, the closure of the group generated by all the rotations about each of the subspaces equals SO(n). The latter result also shows that a closed set in Rn that is invariant un- der rotations about more than one axis must be a union of spheres with their centers at the origin.